NBER WORKING PAPER SERIES

RISK, AMBIGUITY, AND THE EXERCISE OF EMPLOYEE STOCK OPTIONS

Yehuda IzhakianDavid Yermack

Working Paper 19975http://www.nber.org/papers/w19975

NATIONAL BUREAU OF ECONOMIC RESEARCH1050 Massachusetts Avenue

Cambridge, MA 02138March 2014

We appreciate helpful comments by Steven Huddart and Jennifer Carpenter. The views expressedherein are those of the authors and do not necessarily reflect the views of the National Bureau of EconomicResearch.

NBER working papers are circulated for discussion and comment purposes. They have not been peer-reviewed or been subject to the review by the NBER Board of Directors that accompanies officialNBER publications.

© 2014 by Yehuda Izhakian and David Yermack. All rights reserved. Short sections of text, not toexceed two paragraphs, may be quoted without explicit permission provided that full credit, including© notice, is given to the source.

Risk, Ambiguity, and the Exercise of Employee Stock OptionsYehuda Izhakian and David YermackNBER Working Paper No. 19975March 2014JEL No. G12,G13,G34,J33

ABSTRACT

We investigate the importance of ambiguity, or Knightian uncertainty, in executives’ decisions aboutwhen to exercise stock options. We develop an empirical estimate of ambiguity and include it in regressionmodels alongside the more traditional measure of risk, equity volatility. We show that each variablehas a statistically significant effect on the timing of option exercises, with volatility causing executivesto hold their options longer in order to preserve remaining option value, and ambiguity increasingthe tendency for executives to exercise early in response to risk aversion. Regression estimates forthe volatility and ambiguity variables imply similar magnitudes of economic impact upon the exercisedecision, with the volatility variable being about 2.5 times stronger.

Yehuda IzhakianZicklin College of BusinessOne Bernard Baruch WayNew York, NY 10010yud@stern.nyu.edu

David YermackStern School of BusinessNew York University44 West Fourth Street, Suite 9-160New York, NY 10012and NBERdyermack@stern.nyu.edu

2

Risk, Ambiguity, and

the Exercise of Employee Stock Options

I. Introduction

This paper explores the role of Knightian uncertainty, also known as ambiguity, in executives’

decisions about when to exercise their stock options. Our results should help clarify a

longstanding division (or ambiguity) in the compensation literature over whether risk is

positively or negatively associated with the early exercise of executive options. Two competing

hypotheses predict that risk may cause executives to exercise stock options early, due to their

impulse to diversify, or alternatively, that risk may cause options holders to delay exercise, since

the remaining option value increases when risk is high. Table 1 lists six major studies of

employee stock option exercises in recent years, using very large datasets with observations

ranging between 1985 and 2009. These studies are decidedly split on whether greater risk, as

measured by equity volatility, predicts earlier or later exercise of options.

In our regression model of option exercises, we introduce the concept of ambiguity

alongside the more widely used measure of equity volatility, because each of these measures of

risk is potentially important to employee option holders. Ambiguity is based upon the concept of

Knightian uncertainty, introduced in Knight’s (1921) treatise on Risk, Uncertainty, and Profit.

Knightian uncertainty refers to conditions under which the set of events that may occur are a

priori unknown, and the odds of these possible events are also either not unique or are unknown.

Knightian uncertainty is distinctly different than volatility and related measures of financial risk,

which typically treat the set of future outcomes (such as the range of future stock prices) as

known, with certain probabilities assigned to them and adding to one. The most familiar public

3

discussion of Knightian uncertainty probably occurred in 2002, when U.S. Secretary of Defense

Donald Rumsfeld drew a distinction between “known unknowns” and “unknown unknowns”

when discussing hypothetical chemical weapons held by Iraq.

Previous studies have focused on the theoretical aspect of ambiguity, while not testing

their models empirically. In general these studies employ decision models that do not provide

separation between risk and ambiguity and between preferences and beliefs. Therefore, they do

not provide a straightforward way for measuring ambiguity.1 We believe our study is the first to

introduce ambiguity into an option pricing model, and we use an empirical method to estimate

the impact of ambiguity upon the timing of stock option exercises while controlling for the

impact of volatility, the more traditional measure of risk. We find that ambiguity and volatility

measure distinctly separate aspects of financial uncertainty, as they have a simple correlation that

is weakly positive but not far from zero.

Our analysis provides consistent evidence that both volatility and ambiguity are

significantly associated with option exercises, but in opposite directions. Our regression

estimates indicate that high ambiguity is associated with earlier option exercises, while high

volatility is associated with later exercises. Ambiguity therefore appears to influence exercise

policy through the channel more commonly described as risk aversion. The magnitudes of the

coefficient estimates for the two variables imply that they operate with very similar economic

strength in executives’ exercise decisions, with the volatility variable appearing to have about 2.5

times as much influence as the ambiguity variable.

Our results add to a growing literature of papers that fit models of stock option exercise

using predictor variables beyond those that appear in the classical binomial or Black-Scholes

1 Earlier literature dealt with this limitation either by calibration (for example, Ju and Miao (2011)) or by attributing

the disagreement of professional forecasters to ambiguity (for example, Anderson et al. (2009)).

4

models. In addition to the papers listed in Table 1, an interesting related paper is Spalt (2013),

which studies awards rather than exercises of executive stock options. Like us, the author of that

paper uses a framework based on cumulative prospect theory to estimate employees’ perceived

value of option awards, with the results of the paper hinging upon a transformation of

probabilities of future realizations of the underlying stock price such that employees’ beliefs

deviate from the true distribution. The author finds that firms with high idiosyncratic volatility

grant more options to their employees, a result that can be supported by prospect theory but not

by classical option pricing models.

The remainder of this paper is organized as follows. Section II presents a theoretical

discussion of ambiguity and develops our hypotheses. Section III describes our sample selection

and variable definitions, including the estimates of the ambiguity and volatility variables that are

central to our investigation. Section IV presents regression analysis of option exercise, and

Section V concludes the paper.

II. Ambiguity

Many real-life financial decisions involve ambiguity. As a simple example, suppose two urns

contain a mixture of black and yellow balls in unknown and possibly different proportions, and

you are offered a prize for selecting a yellow ball with only one attempt. The first urn contains

30 balls, and the second urn contains 10. From which urn should you choose to make your

selection?

Treating ambiguity analytically can help decision-makers to rank order alternatives such

as the two urns in the game above. With employee stock options, the decision above is similar to

5

the decision of whether to continue holding an option or to exercise it when the degree of

ambiguity changes.

II.1. The decision theoretic model of ambiguity

Knightian uncertainty has provided the basis for a rich literature in decision theory that begins

with Schmeidler (1989) and Gilboa and Schmeidler (1989). These pioneering studies provide

axiomatic foundations for modeling decision making under ambiguity, and they suggest that in

the presence of ambiguity, an individual's beliefs may take the form of either multiple priors or a

single but nonadditive prior. In their model of max-min expected utility with multiple priors,

Gilboa and Schmeidler (1989) assert that an ambiguity-averse individual possesses a set of priors

and evaluates her ex ante welfare conditional upon the worst prior. The subjective nonadditive

probabilities of Gilboa (1987), the Choquet expected utility of Schmeidler (1989) and the

cumulative prospect theory of Tversky and Kahneman (1992) suggest that, in the presence of

ambiguity, the probabilities that reflect the individual's willingness to bet might not be additive.

Earlier literature on ambiguity focuses mainly on its theoretical aspects, with special

attention to the implications of ambiguity for the equity premium. See, for example, Cao et al.

(2005), Nau (2006), Epstein and Schneider (2008) and Izhakian and Benninga (2011). Other

studies attempt to provide explanations for puzzling asset pricing phenomena by introducing

ambiguity into conventional asset pricing models. For example, Pflug and Wozabal (2007) and

Boyle et al. (2011) suggest extensions of the mean–variance approach to incorporate ambiguity.

Additional papers have developed extensions of the capital asset pricing model that incorporate

ambiguity, such as Chen and Epstein (2002), Maccheroni et al. (2009) and Izhakian (2012b).

6

We distinguish the concepts of risk and ambiguity by using the theoretical framework of

expected utility with uncertain probabilities (EUUP). EUUP, proposed by Izhakian (2012a), is

based on Schmeidler's (1989) Choquet expected utility but puts more structure on subjective

probabilities (capacities) and adds reference-dependent beliefs. Like Tversky and Kahneman's

(1992) cumulative prospect theory, EUUP assumes that investors have a reference point relative

to which outcomes are classified as unfavorable (a loss) or as favorable (a gain). Outcomes

below than the reference point are considered unfavorable, and higher outcomes are considered

favorable. The cumulative probability of unfavorable outcomes plays an important role in

estimating the degree of ambiguity. Unlike cumulative prospect theory, EUUP does not assume

different attitudes toward risk for losses and for gains (i.e., loss aversion).

EUUP assumes that a random variable is subject to two tiers of uncertainty, one with

respect to the value of outcomes, and the other with respect to the probabilities of these

outcomes. Each tier of uncertainty is modeled by a separate state space. This structure

introduces a complete distinction of risk from ambiguity with regard to both beliefs and

preferences. The degree of risk and investors' attitudes toward it are then measured with respect

to the first space, while ambiguity and investors' attitudes toward it are measured with respect to

the second space. In our approach, the separability of risk and ambiguity represents an

assumption. While previous literature (e.g., Schmeidler (1989) and Gilboa and Schmeidler

(1989)) makes a considerable contribution to understanding preferences with respect to

ambiguity, a complete separation between ambiguity and risk has not yet been derived.

Formally, let be a probability space, where is an uncertain (random)

probability measure, and the set of probability measures is closed and convex. The set is

equipped with a Borel probability measure, denoted , with a bounded support. Given a random

7

variable, its random mean and random variance , conditional upon the

random probability measure , are denoted by the Greek letters and , respectively.

Investors have distinct preferences concerning risk and concerning ambiguity. As usual,

preferences concerning risk are modeled by a strictly-increasing, continuous and twice-

differentiable utility function , which in our case is normalized to . Risk

aversion takes the form of a concave , risk loving the form of a convex , and risk neutrality

the form of a linear . Preferences concerning ambiguity are modeled by a strictly-increasing,

continuous and twice-differentiable function over probabilities, [- ] , called the outlook

function. Similarly to risk, ambiguity aversion takes the form of a concave , while ambiguity

loving takes the form of a convex , and ambiguity neutrality the form of a linear . In EUUP,

ambiguity aversion occurs when an investor prefers the expectation of an uncertain probability of

each outcome over the uncertain probability itself.2

Because individuals may interpret random probabilities in nonlinear ways, perceived

probabilities in EUUP are nonadditive, meaning that they do not necessarily add to 1.

Ambiguity aversion results in a subadditive probability measure, while ambiguity loving results

in a superadditive measure. The expected utility of consuming the future risky and ambiguous

outcome of some investment is, therefore, formed by:

∫ (∫ ( )

)

∫ (∫ ( )

)

, (1)

where is a reference point that distinguishes between unfavorable outcomes (losses)

and favorable outcomes (gains). The function , proposed by Izhakian (2012a), is based upon the

2 Recall that risk aversion is exhibited when an investor prefers the expected return of the random return over the

random return itself.

8

functional representation of Wakker (2010) and Kothiyal et al. (2011). This function applies a

two-sided Choquet integration to gains and to losses, relative to a reference point . When

investors are ambiguity neutral, i.e., is linear, Equation (1) collapses to the conventional

expected utility.

II.2. Implications of ambiguity for option pricing

To study the impact of ambiguity on option prices, consider a binomial asset with the outcomes

and in the bad and the good states of nature, respectively. Assume that the reference point

satisfies [ ] , where [ ] is the expected outcome taken with respect to the expected

probabilities [ ] ∫

. By Equation (1), the value of this asset in terms of

expected utility is

(∫ ( )

) (∫ ( )

) . (2)

To observe the distinct impact of ambiguity and preference concerning it on the asset

value, one can take a second Taylor approximation of with respect to the uncertain

probabilities around the expected probability of each event separately, to obtain

( [ ]

[ ]

[ ] [ ])

( [ ]

[ ]

[ ] [ ])

. (3)

Since every probability measure is additive, [ ] [ ] Using this notion

of variance of probabilities, the degree of ambiguity can be measured by

[ ] [ ] [ ] (4)

9

where [ ] stands for the variance of the cumulative probability of unfavorable events,

and [ ] stands for the variance of the cumulative probability of favorable events, taken

with respect to the second order probabilities The measure ( ) can be used in the

general case of random variables with multi states of nature, see Izhakian (2012a).3

Ambiguity and aversion to it are modeled in Equation (3) through the investor’s

perceived probabilities. Consider, for example, the good outcome. The expression

[ ]

[ ]

[ ] [ ] (5)

is the perceived probability of this outcome occurring. This probability is a function of the

degree of ambiguity, measured by [ ], and the investors’ preferences concerning

ambiguity, formed by the coefficient of absolute ambiguity aversion

. Aversion to ambiguity,

modeled by a concave implies

. For an ambiguity-averse investor, a higher aversion to

ambiguity or a higher degree of ambiguity result in lower perceived probabilities, which in turn

imply lower expected utility.

Consider now a one-period call option on the binomial asset , with the exercise price

. By Equation (3) the value of this option (in terms of expected utility) is

( [ ]

[ ]

[ ] [ ]) . (6)

By this equation we can make the following claims:

Claim 1: The option value increases with the risk of the underlying equity.

3 In asset pricing the degree of ambiguity can be decomposed into systematic and idiosyncratic ambiguity. See

Izhakian (2012b).

10

To see this, assume that the risk of the underling increases such that the bad outcome is

and the good outcome is , for some Clearly, by Equation (5), the value of

the call option increases in risk.

Claim 2: The option value decreases with the aversion to ambiguity.

Higher aversion to ambiguity implies a lower [ ]

[ ] , lower perceived probabilities,

and therefore a lower value of the option. To see this, take for example an ambiguity preference

of the constant absolute ambiguity aversion type. In this case [ ]

[ ] - , where is the

coefficient of absolute ambiguity aversion. Higher aversion to ambiguity implies a higher and

therefore a lower - .

Claim 3: Assuming ambiguity-averse investors, the option value decreases with the

ambiguity of the underlying equity.

Since aversion to ambiguity implies a negative [ ]

[ ] , a higher ambiguity of the underlying

equity, measured by [ ] implies a lower perceived probabilities (Equation (5)) and therefore a

lower value of the option.

II.3. Binomial example

In this section we illustrate the impact of ambiguity and risk on option prices with a simple

numerical example. Consider an underlying equity whose current price is $1. In the next period

its price can be either $1.1 or $0.9, i.e., respectively +10% or -10% returns. Assume an

ambiguity-averse investor of a constant absolute ambiguity aversion type with coefficient of

11

ambiguity aversion -

. For simplicity, assume that the investor is risk neutral and has

a reference point . Recall that the reference point distinguishes between losses and gains.

Let a one period call option have an exercise price and assume that the

probabilities of each, the bad outcome and the good outcome, are exactly 50%, which means that

no ambiguity is present. In this case, the variance of the probability of the favorable outcome is

0 and therefore, by substituting into equation (5), the value of the call option is is

. If the risk of the underlying equity increases such that the bad and the good

outcomes are, respectively, $0.8 and $1.2, then the value of the call option increases to

.

Assume now that the probabilities of the future outcomes of the underlying equity are

ambiguous such that the outcomes are distributed either or with equal

likelihood. In this case, the expected probability of the favorable outcome is [ ]

and its variance is [ ]

, implying that the degree of ambiguity is Substituting into

Equation (5), the value of the call option (in terms of expected utility) is ( -

) For an investor with higher aversion to ambiguity, say the value of

the option (in terms of expected utility) drops to ( -

) If

the degree of ambiguity of the underlying equity increases such that the future outcomes are

distributed either either or , then the expected probability of the favorable

outcome remains unchanged [ ] but the variance of its

probability increases to [ ] ( - ) ( - )

, implying a

12

degree pf ambiguity of Substituting into Equation (5), the value of the

call option drops to (

)

II.4. Hypotheses

We turn now to developing our hypotheses based upon the claims of Section II.2. In particular,

Claim 1 proposes that a higher volatility of the underlying equity implies a higher value of the

option. On the other hand, Claim 3 proposes that a higher ambiguity of the underlying equity

implies a lower value of the option. These claims suggest that the risk and the ambiguity

associated with the underlying equity have opposite effects on the incentive that employees have

to exercise their vested options.

Hypothesis H1: When the expected risk associated with the underlying asset is high, employees

tend to hold their options rather than exercise them, and vice versa.

At each interval of time, an option holder has to decide whether to exercise his vested

options or keep the option alive and defer the decision to the next period. His choice is based

upon the future expected value of the option relative to its current value, assuming that the

market price of the underlying equity reflects its value in terms of expected utility. If the option

holder expects the volatility of the underlying equity to be high in the next month, then he

anticipates a higher value of option in the next month. Therefore, to maximize his expected

utility, he would prefer to continue holding the option and leave it unexercised.

Hypothesis H2: When the expected ambiguity associated with the underlying asset is high,

employees tend to exercise their options rather than hold them, and vice versa.

13

Claim 3 demonstrates that ambiguity and option value are negatively related. This means

that if the ambiguity of the underlying asset is expected to be high, the value of an option on this

equity is expected to be low.

III. Data description

III.1. Sample selection

We estimate the relation between ambiguity and the timing of option exercises using a sample

from Table 2 of the Thomson Reuters Insiders Data database. After a process of data cleaning

and filtering described below, we analyze 63,244 option exercises by 19,192 employees in 3,006

individual firms. Our unit of observation for regression analysis is an award-month, and we

include each month for which an option is vested (or exercisable) and not yet fully exercised,

2,142,732 observations in all.

To construct our sample, we begin with a download of 5,322,257 records from the entire

Thomson Reuters database from January 1996 to December 2013, with each record representing

either an award or an exercise of a given grant to a given employee in a specific firm. Each

unique grant is identified using the firm identifier, security identifier, person identifier, vesting

date and termination date. When one of these descriptors is missing, we attempt to complete it

using information from other records of the same grant. After this data cleaning, we drop all

records missing one of the following descriptors: vesting date, expiration date, number of

options, or exercise price. In addition, we drop all records that we cannot match with identifiers

to the CRSP stock price database. This filtering leaves us with 3,862,174 records. After

dropping duplicate records, 3,552,491 are left.

14

Next, we aggregate the exercise and cancellation transactions related to each grant on a

monthly basis. All grants that are never exercised are dropped (many may have expired out of

the money, some are still active today, and others may have been exercised after the option

holder left the company and had no reporting obligation). We identify 107,930 unique option

grants for which we have both award and exercise data. For each month for each of these grants,

we construct a cumulative total of the number of vested options that have not yet been exercised.

When the database reports an exercise quantity equal to or exceeding the number of options held,

we consider the observation to represent an exercise of all of the remaining options and drop all

future months. Using this method, we are left with a sample of 4,007,854 monthly observations

for the 107,930 unique grants, representing the period between the vesting date and the exercise

date for each grant.

We reduce this sample of 4,007,854 observations to a sample suitable for regression

analysis after calculating the necessary independent variables for each observation. In general

we require 12 months of stock price history to construct measures of expected volatility and

expected ambiguity from daily and intraday returns as described below. This was possible for

2,586,265 observations. Finally, we drop all out-of-the-money options, based upon the closing

price at the end of the prior month. This leaves us with 2,142,732 monthly observations during

which employees could have exercised their options. Our main dependent variable is the

percentage of options exercised by the option holder in a particular month, equal to the number

exercised over the number held at the start of the month. Because option exercises occur only

infrequently, for most months this variable equals zero. This property of the data suggests that a

Tobit estimation would be most appropriate, but it was not possible for us to fit a Tobit model

with fixed effects and clustered standard errors. Therefore, we a general linear mixed model

15

(GLMM) with clustering and fixed effects in the regression analysis below. In unreported

sensitivity tests, we find little difference between our reported GLMM estimates and Tobit

estimates without fixed effects and clustered standard errors.

III.2. Estimating expected ambiguity

To estimate the degree of ambiguity of a given underlying asset from trading data, one must first

elicit the possible probabilities of unfavorable outcomes (returns). To do so, one must make

assumptions about the nature of the probability distribution of returns and the investors’

reference point. We define the reference point to be the risk-free rate of return, denoted . That

is, any return lower than is considered an unfavorable and any return greater than is

considered favorable (alternatively, one could take 0 to be the reference point). In addition we

assume that returns on assets are log-normally distributed; therefore, the degree of ambiguity of

the return on the underlying asset can be measured by

[ ] [ ( )], (7)

where ( )

√ ∫

( )

is the cumulative probability of unfavorable

outcome and and are the random mean and the random variance of the return on the

underlying equity, respectively.

We employ the empirical method developed by Brenner and Izhakian (2011, 2012) to

estimate the degree of ambiguity using intraday stock trading data from the TAQ database. We

compute the degree of ambiguity given in Equation (7), for each stock and for each month, by

applying the following procedure. We sample the price of the stock every five minutes starting

from 9:30 until 16:00. In cases where there is no trade at a specific time interval, we take the

16

volume-weighted average of the closest trading price. Using these prices we compute five-

minute returns, 78 times for each day. Note that this procedure implies that we ignore returns

between each closing price and opening price on the following day, thereby eliminating the

impact of overnight price changes and dividend distributions. For each stock, we drop all trading

days that do not have at least 15 quotes representing different five minute time intervals, and we

drop all trading months that do not have at least 15 days that satisfy this condition. Observations

with extreme price changes (minus or plus 20 percent log returns) within five minutes are also

dropped.

For a given stock, for each day we compute the mean and variance of five-minute returns

applying the adjustment for non-synchronous trading proposed by Scholes and Williams (1977)

and a correction for heteroscedasticity. Then, by the assumption that the intraday returns are log-

normally distributed, we compute for each day the probability of an unfavorable outcome (loss)

using the cumulative normal probability distribution. For each month, there are 20 to 22

different loss-probabilities. From these loss-probabilities we compute the standard deviation of

the probability of loss to obtain the degree of ambiguity, , for an individual stock in a given

month. The average ambiguity (over firms) is positively skewed. To adjust for skewness, we

examine the natural logarithm of . Finally, we assume that investors form an expectation

about future ambiguity by taking into account the evolution of a given stock’s ambiguity over

time. Regression analysis indicates that the average autocorrelations (over firms) of ( )

decays beyond lag 9, suggesting that ( ) follows a nine-order autoregressive process.

Based upon the Akaike information criterion (AIC), for each equity , we estimate the

conditional ambiguity using an AR(7) with one period autocorrelated residuals model of the

17

time-series of the natural logarithm of ambiguity. For every month we perform the following

time-series regression for each equity :

( ) ( ) ( ) . (7)

Then we estimate the expected ambiguity by

( ( )

( )), (8)

where ( ) is estimated using coefficients estimated by Equation (7).

III.3. Estimating expected volatility

Along with ambiguity, volatility serves as the most important explanatory variable in our

analysis. We compute volatility with standard methods, using daily log returns adjusted for

dividends obtained from the CRSP database. For each individual stock in a given month we

calculate the standard deviation, of the stock’s daily returns over that month, again applying

the Scholes and Williams (1977) correction for non-synchronous trading and a correction for

heteroscedasticity.4 As with ambituity, the average volatility measures (over firms) are also

positively skewed. To adjust for skewness we examine the natural logarithm of . Again we

assume that investors form an expectation about future volatility based upon its evolution over

time for a given stock. The average autocorrelations (over firms) of ( ) decay beyond lag 6,

suggesting that ( ) follows a six-order autoregressive process. Examining the cross

correlations between the volatility of equity and its ambiguity in lags - - , suggests

that, on average, the ambiguity the preceding three months has a significant impact on volatility.

Based upon the AIC, for each equity in every month we estimate conditional volatility by AR(6)

with two periods autocorrelated residuals model of the time-series of the natural logarithm of

4 See, for example, French, Schwert and Stambaugh (1987).

18

volatility. Similar to the procedure applied to ambiguity, for every month we perform the

following time-series regression for each equity:

( ) ( ) ( ) , (9)

and we then estimate the expected volatility by

( ( )

( )), (10)

where ( ) is estimated using coefficients estimated by Equation (9).

III.3. Other explanatory variables

In addition to expected ambiguity and expected volatility, our models of option exercises include

a number of standard control variables, many of them based on the prior studies listed in Table 1.

We include two indicators related to behavioral finance predictions. One indicator equals one

during the vesting month, representing the first month in which a given option award can be

exercised, and we expect this to have a positive association with the probability of exercise. The

other indicator equals one if the stock hits a 12-month high during the prior month, and we also

expect this variable to have a positive association with the probability of exercise. The log of

remaining option life (measured as one plus the number of months to expiration) should exhibit a

negative relation with the probability of exercise, based on standard comparative statics of option

value with respect to time, while the dividend yield of the underlying stock should exhibit a

positive relation for the same reason. Following Carpenter, Stanton, and Wallace (2013), we

multiply the dividend yield times an indicator that equals one if a dividend is payable in the

current month. We include the log of the ratio of prior month-end stock price divided by the

option exercise price to capture the depth in-the-money of each option award, and we expect this

variable to exhibit a positive association with the probability of exercise. In certain regressions

19

we use the change in ambiguity and change in volatility as explanatory variables, and these are

calculated as the log of the ratio of the current month’s value over the prior month’s value.

Table 2 presents summary statistics for the key variables used in the paper, and Table 3

presents a correlation matrix for these variables. The two measures of risk, expected volatility

and expected ambiguity, are only weakly related to one another, with an estimated correlation of

0.186. This suggests that the two variables capture economically distinct aspects of financial

uncertainty.

IV. Empirical results

Our regression analysis of how risk impacts executives stock option exercises appears in Table 4.

As described above, the sample for analysis includes more than 2.14 million award-month

observations in which an executive holds vested, in-the-money options at the start of the month

and faces a decision of whether to exercise the option or continue it for another month. The

dependent variable in all models is the percentage of an award that is exercised in a given month.

All models include standard errors clustered by person, firm, month, and year in order to reflect

the lack of independence of multiple observations within executives and within periods of time.

The first two columns of Table 4 show estimates for very basic models with only an

intercept and the volatility variable, in the first column, and an intercept and the ambiguity

variable, in the second column. Both risk variables appear in a model with an intercept in the

third column, and their estimates change little when they are included in the same regression

alongside one another. The volatility variable has a negative and significant estimate of

approximately -0.15, while the ambiguity variable has a positive and significant estimate of

approximately +0.02. Moving from left to right in Table 4, we introduce control variables as

20

described above and fixed effects for years, persons, and firms. Finally, in the right column of

Table 4, we show estimates for the most general model, with all control variables and fixed

effects included. Both the volatility and ambiguity variables have significant estimates in every

column except one, and they remain relatively unchanged when control variables are added.

Our results imply that risk impacts the option exercise decision in a multi-faceted way, as

we find that the volatility and ambiguity variables influence exercises in opposite directions.

Our negative and significant estimates for the volatility variable are similar to those found in the

recent study by Carpenter, Stanton, and Wallace (2013), the only one of the six papers listed in

Table 1 that finds this pattern unambiguously. The Carpenter et al. study appears to have several

advantages compared to the others in the table, including a much larger sample over a far longer

time period, as well as a more sophisticated estimation method. However, none of the papers

considers measures of uncertainty other than the traditional volatility variable, and one of the

contributions of our paper appears to be our illustration that different types of uncertainty impact

financial decisions sometimes in opposing directions.

To assess the economic significance of the two variables’ estimates, we multiply the

coefficients in the right column of Table 4 by the standard deviation of each variable as shown in

Table 2. The product represents the impact on the exercise decision of a one standard deviation

change of each variable. For the volatility variable the relevant calculation is

or about 0.18%, and for the ambiguity variable the calculation is , or

about 0.07%, implying that the equity volatility variable as about two and a half times stronger

influence upon executives’ exercise decisions. The unconditional mean of the dependent

variable, as shown in Table 2, is about 2.54%. The two uncertainty variables therefore appear to

21

have meaningful, but somewhat modest, impact on executives’ month-to-month decisions about

whether to exercise their stock options.

Other estimates for variables in Table 4 follow the expected pattern found in other

papers. Managers are significantly more likely to exercise their stock options soon after the

underlying shares reach a 12-month high, and also in the first month after vesting. While the

former result seems to be rooted in behavioral finance, the latter may be an artifact of certain

executives planning to change jobs but waiting until their options vest and can be liquidated. We

find that options are more likely to be exercised when they are deep in-the-money, when large

dividends are payable in the current month, and when the time to expiration grows shorter. All

three of these results square with standard comparative statics predictions.

In tables 5 and 6 we present a series of robustness tests to show the consistency of our

estimates for the two uncertainty variables. Table 5 displays estimates for regressions with the

same specification as the right column of Table 4, in which we include the full range of control

variables and fixed effects, alongside three alternative specifications of the volatility and

ambiguity variables. We tabulate only the intercept and two uncertainty variables’ coefficients

in order to save space and highlight the important results. In the first column of Table 5, we

reproduce the estimates from the right column of Table 4 for comparison purposes. In the

second column of the table, the two uncertainty variables are measured one month earlier, and in

the third column, they are measured with two months’ lag. Finally, in the right column we take

the first difference of each uncertainty variable, equal to the value at the close of the immediately

preceding month, minus the value one month earlier. In all four columns of Table 5, the

estimates for the uncertainty variables have the same sign, significance, and approximate

22

magnitude, although the coefficients are noticeably closer to zero in the right column when we

use the first-difference specification.

In Table 6, we continue our robustness tests by fitting our regression model over certain

subsets of observations. In the first column, we look at observations for chief executive officers

only, according to their job titles reported in the Thomson Reuters database. These option

exercises comprise about 16% of the overall sample. In the center column, we estimate our

regression for those observations in the top quartile of the volatility variable, and in the right

column, we use those observations that lie in the top 25% of ambiguity. All models contain the

full range of control variables and fixed effects used in the preceding table. Once again we find

a high degree of similarity for the uncertainty variables’ coefficient estimates, as each variable

retains its sign, significance, and approximate magnitude within each of these subsamples of

observations.

V. Conclusion

We investigate the importance of ambiguity, or Knightian uncertainty, in executives’ decisions

about when to exercise stock options. The impact of risk on the timing of option exercise has

motivated several prior studies which have failed to reach a conclusion about whether the

relation is positive or negative. Our contribution involves the introduction of a second measure

of uncertainty, ambiguity, alongside the more traditional measure of volatility. The empirical

estimates of these two quantities exhibit only a modest correlation, and both turn out to be

important. We show that each variable has a statistically significant effect on the timing of

option exercises, with volatility causing executives to hold their options longer in order to

preserve remaining option value, and ambiguity increasing the tendency for executives to

23

exercise early in response to risk aversion. Regression estimates for volatility and ambiguity

variables imply very similar magnitudes of economic impact upon the exercise decision, with the

volatility variable appearing to impact about 2.5 times more strongly than the ambiguity variable.

24

References

Anderson, E. W., E. Ghysels, and J. L. Juergens (2009) “The Impact of Risk and Uncertainty on

Expected Returns," Journal of Financial Economics 94, 233-263.

Armstrong, C., A. D. Jagolinzer, and D. F. Larcker (2007) “Timing of Employee Stock Option

Exercises and the Cost of Stock Option Grants,” Stanford University working paper, available at

www.ssrn.com/abstract=905280.

Bettis, J., J. Bizjak, and M. Lemmon (2005) “Exercise Behavior, Valuation, and the Incentive

Effects of Employee Stock Options,” Journal of Financial Economics 76, 445-470.

Boyle, P., L. Garlappi, R. Uppal, and T. Wang (2011) “Keynes Meets Markowitz: The Tradeoff

Between Familiarity and Diversification," Management Science 57, 1-20.

Brenner, M., and Y. Izhakian (2011) “Asset Prices and Ambiguity: Empirical Evidence," New

York University working paper, available at www.ssrn.com/abstract=2284649.

Brenner, M., and Y. Izhakian (2012) “Pricing Systematic Ambiguity in Capital Markets," New

York University working paper, available at www.ssrn.com/abstract=2119040.

Cao, H. H., T. Wang, and H. Zhang (2005) “Model Uncertainty, Limited Market Participation,

and Asset Prices," Review of Financial Studies 18, 1219-1251.

Carpenter, J., R. Stanton, and N. Wallace (2013) “The Importance of Behavioral Factors in the

Exercise of Employee Stock Options,” New York University working paper, available at

people.stern.nyu.edu/jcarpen0/pdfs/CarpenterStantonWallace2013.pdf.

Chen, Z., and L. Epstein (2002) “Ambiguity, Risk, and Asset Returns in Continuous Time,"

Econometrica 70, 1403-1443.

Epstein, L. G., and M. Schneider (2008) “Ambiguity, Information Quality, and Asset Pricing,"

Journal of Finance 63, 197-228.

French, K. R., G. W. Schwert, and R. F. Stambaugh (1987) “Expected stock returns and

volatility," Journal of Financial Economics, Vol. 19, No. 1, pp. 3-29.

Gilboa, I. (1987) “Expected Utility with Purely Subjective Non-Additive Probabilities," Journal

of Mathematical Economics 16, 65-88.

Gilboa, I., and D. Schmeidler (1989) “Maxmin Expected Utility with Non-Unique Prior,"

Journal of Mathematical Economics 18, 141-153.

Hemmer, T., S. Matsunaga, and T. Shevlin (1996) “The Influence of Risk Diversification on the

Early Exercise of Employee Stock Options by Executive Officers,” Journal of Accounting and

Economics 21, 45-68.

25

Huddart, S., and M. Lang (1996) “Employee Stock Option Exercises: An Empirical Analysis,”

Journal of Accounting and Economics 21, 5-43.

Izhakian, Y. (2012a) “A Theoretical Foundation of Ambiguity Measurement," New York

University working paper, available at people.stern.nyu.edu/yizhakia/papers/ambgmsr.pdf.

Izhakian, Y. (2012b) “Capital Asset Pricing under Ambiguity," New York University working

paper, available at www.ssrn.com/abstract=2007815.

Izhakian, Y., and S. Benninga (2011) “The Uncertainty Premium in an Ambiguous Economy,"

Quarterly Journal of Finance 1, 323-354.

Ju, N., and J. Miao (2012) “Ambiguity, Learning, and Asset Returns," Econometrica 80, 559-

591.

Klein, D., and E. Maug (2011) “How Do Executives Exercise Their Stock Options?” University

of Mannheim working paper, available at ssrn.com/abstract=1342316.

Knight, F. (1921) Risk, Uncertainty, and Profit (Boston: Houghton Mifflin).

Kothiyal, A., V. Spinu, and P. P.Wakker (2011) “Prospect Theory for Continuous Distributions:

a Preference Foundation," Journal of Risk and Uncertainty 42, 195-210.

Maccheroni, F., M. Marinacci, A. Rustichini, and M. Taboga (2009) “Portfolio Selection with

Monotone Mean-Variance Preferences," Mathematical Finance 19, 487-521.

Nau, R. F. (2006) “Uncertainty Aversion with Second-Order Utilities and Probabilities,"

Management Science 52, 136-145.

Pug, G., and D. Wozabal (2007) “Ambiguity in Portfolio Selection," Quantitative Finance 7,

435-442.

Schmeidler, D. (1989) “Subjective Probability and Expected Utility without Additivity,”

Econometrica 57, 571-587.

Scholes, M., and J. Williams (1977) "Estimating Betas from Nonsynchronous Data," Journal of

Financial Economics 5, 309–327.

Spalt, O. (2013) “Probability Weighting and Executive Stock Options,” Journal of Financial and

Quantitative Analysis 48, 1085-1118.

Wakker, P. (2010) Prospect Theory: For Risk and Ambiguity (Cambridge, UK: Cambridge

University Press).

26

Table 1

Previous studies of risk and executive stock option exercises

The table shows six previous studies of executive stock option exercises, with information about the sample period, sample size,

measurement of risk, and the estimated impact of risk upon early option exercise.

Study

Sample

period

Sample size

Measurement of

volatility

Impact of risk on

early exercise

Remarks

Huddart &

Lang (1994)

c. 1985-

1995

85,853 exercises by

58,316 employees (all

levels) at 8 companies

Daily stock returns

over one year prior to

exercise month

Mixed Estimates vary across companies

and job ranks and are sensitive to

estimation method

Hemmer,

Matsunaga &

Shevlin (1996)

1990 110 exercises by 74

officers and directors at

65 firms

Monthly stock returns

over five years prior

to 1990

Positive Volatility is scaled by Black-

Scholes hedge ratio

Bettis, Bizjak

& Lemmon

(2005)

1996-

2002

141,020 exercises by

officers and directors at

3,966 companies

Monthly stock returns

over three years prior

to exercise date

Positive

Armstrong,

Jagolinzer &

Larcker (2007)

c. 1995-

2005

17,570 exercises by

15,409 employees (all

levels) at 10 firms

Daily stock returns

over one year prior to

exercise day

Not significant

Klein & Maug

(2011)

1996-

2008

23,646 exercises by

13,948 officers and

directors at 2,008 firms

Weekly stock returns

over one year prior to

vesting date

“Weak but

positive”

Estimates are significant only in

some models

Carpenter,

Stanton &

Wallace (2013)

1981-

2009

687,594 exercises by

419,822 employees at

102 firms

Daily stock returns

over 66 days prior to

grant date

Negative Uses computationally intensive

GMM estimation

27

Table 2

Descriptive statistics

The table shows descriptive statistics for the key dependent and explanatory variables in the regression analysis of option exercise

timing. The sample includes 2.14 million monthly observations associated with 63,244 option grants that are exercised by 19,192

employees in 3,006 individual firms between 1996 and 2013, using records from the Thomson Reuters Insiders Data file. For a given

option award, the percentage of options exercised is the number exercised in a given month divided by the number of vested options

held at the start of the month. Expected volatility, based on CRSP daily stock price records, and expected ambiguity, based on TAQ

intra-day stock price records, are calculated according to procedures described in the text.

Variable

Observations Mean Std. Dev. Minimum Median Maximum

Percentage of options exercised

2,142,732 0.0254 0.1532 0 0 1

Expected volatility

2,142,732 0.0176 0.0078 0.0004 0.0161 0.1950

Expected ambiguity

2,142,732 0.4954 0.0522 0.1129 0.4970 1.5159

Vesting month indicator

2,142,732 0.0191 0.1369 0 0 1

Log (1+months to expiration)

2,142,732 7.2887 0.8202 0.6931 7.5256 9.3357

12-month high price indicator

2,142,732 0.0568 0.2315 0 0 1

Dividend yield

dividend month indicator

2,142,732 0.0002 0.0020 0 0 0.0954

Log (stock price / exercise price)

2,142,732 0.8753 0.7505 0.0000 0.7177 14.1998

28

Table 3

Correlation matrix

The table displays simple correlations between the key dependent and explanatory variables. The sample includes 2.14 million

monthly observations associated with 63,244 option grants that are exercised by 19,192 employees in 3,006 individual firms between

1996 and 2013, using records from the Thomson Reuters Insiders Data file. For a given option award, the percentage of options

exercised is the number exercised in a given month divided by the number of vested options held at the start of the month. Expected

volatility, based on CRSP daily stock price records, and expected ambiguity, based on TAQ intra-day stock price records, are

calculated according to procedures described in the text. All of the correlation estimates in the table are statistically significant below

the 0.0001 probability level.

Percentage

of options

exercised

Expected

volatility

Expected

ambiguity

Vesting

month

indicator

log

(1 + months

to expiration)

12-month

high price

indicator

Dividend

yield

dividend

month

Expected

volatility

-0.0068

Expected

ambiguity

0.0056 0.1855

Vesting month

indicator

0.0385 0.0140 0.0108

log(1 + months to

expiration)

-0.1146 0.0414 0.0256 0.0837

12-month high

price indicator

0.0065 -0.0261 0.0014 0.0077 0.0090

Dividend yield

dividend month

0.0053 -0.0257 -0.0113 0.0089 -0.0073 0.0001

Log (stock price

/ exercise price)

0.0206 0.1441 0.0795 -0.0116 -0.0675 0.0483 -0.0150

29

Table 4

Regression estimates of option exercise timing

Generalized linear mixed model regression estimates of the percentage of an option award that is exercised in the current month. The

sample includes 2.14 million monthly observations associated with 63,244 option grants that are exercised by 19,192 employees in

3,006 individual firms between 1996 and 2013, using records from the Thomson Reuters Insiders Data file. For a given option award,

the percentage of options exercised is the number exercised in a given month divided by the number of vested options held at the start

of the month. Expected volatility, based on CRSP daily stock price records, and expected ambiguity, based on TAQ intra-day stock

price records, are calculated according to procedures described in the text. t-statistics clustered by person, firm, month and year

appear in parentheses below each coefficient estimate.

30

Intercept

Expected volatility

Expected ambiguity

Vesting month indicator

log(1 + months to expiration)

12-month high price indicator

Dividend yield

dividend month indicator

Log (stock price

/ exercise price)

0.0278

(107.67)

-0.1319

(-9.88)

0.0173

(17.33)

0.0164

(8.17)

0.0179

(17.92)

-0.1576

(-11.60)

0.0208

(10.18)

0.1722

(128.21)

-0.1110

(-8.14)

0.0238

(11.75)

0.0542

(71.21)

-0.0220

(-172.48)

0.0042

(9.35)

0.3194

(6.18)

0.0027

(19.13)

0.1715

(46.14)

-0.0006

(-0.14)

0.0199

(9.79)

0.0522

(68.51)

-0.0212

(-156.06)

0.0037

(8.15)

0.2637

(5.11)

0.0022

(15.67)

0.2221

(124.37)

-0.4311

(-22.13)

0.0126

(5.31)

0.0445

(58.44)

-0.0253

(-183.17)

0.0026

(5.84)

0.2740

(5.23)

0.0048

(25.53)

0.2643

(160.95)

-0.2374

(-12.36)

0.0141

(6.07)

0.0324

(42.79)

-0.0313

(-215.89)

0.0012

(2.62)

0.2637

(5.09)

0.0108

(50.79)

0.2646

(160.88)

-0.2353

(-12.22)

0.0141

(6.04)

0.0321

(42.32)

-0.0313

(-215.87)

0.0011

(2.55)

0.2633

(5.08)

0.0109

(50.82)

Year fixed effects

Firm fixed effects

Person fixed effects

Observations

R2

No

No

No

2,142,732

0.00005

No

No

No

2,142,732

0.00003

No

No

No

2,142,732

0.0001

No

No

No

2,142,732

0.016

Yes

No

No

2,142,732

0.020

No

Yes

No

2,142,732

0.032

No

No

Yes

2,142,732

0.060

Yes

Yes

Yes

2,142,732

0.060

31

Table 5

Regression estimates of option exercise timing

Generalized linear mixed model regression estimates of the percentage of an option award that is

exercised in the current month, as a function of lags and changes in the volatility and ambiguity

variables. The sample includes 2.14 million monthly observations associated with 63,244 option

grants that are exercised by 19,192 employees in 3,006 individual firms between 1996 and 2013,

using records from the Thomson Reuters Insiders Data file. For a given option award, the

percentage of options exercised is the number exercised in a given month divided by the number

of vested options held at the start of the month. Expected volatility, based on CRSP daily stock

price records, and expected ambiguity, based on TAQ intra-day stock price records, are

calculated according to procedures described in the text. The regression models contain all of

the control variables from the model in Table 4 above, and the left column in this table

reproduces the estimates from the right column of Table 4 for comparison purposes. t-statistics

clustered by person, firm, month and year appear in parentheses below each coefficient estimate.

Intercept

Expected volatility

Expected ambiguity

Expected volatility t - 1

Expected ambiguity t - 1

Expected volatility t - 2

Expected ambiguity t - 2

Δ Expected volatility

Δ Expected ambiguity

0.2646

(160.88)

-0.2353

(-12.22)

0.0141

(6.04)

0.2687

(163.41)

-0.2073

(-10.79)

0.0049

(2.08)

0.2677

(163.60)

-0.1942

(-10.18)

0.0066

(2.88)

0.2657

(177.96)

-0.0009

(-1.88)

0.0030

(3.72)

Year fixed effects

Firm fixed effects

Person fixed effects

Observations

R2

Yes

Yes

Yes

2,142,732

0.060

Yes

Yes

Yes

2,142,732

0.060

Yes

Yes

Yes

2,142,732

0.060

Yes

Yes

Yes

2,142,732

0.060

32

Table 6

Regression estimates of option exercise timing

Generalized linear mixed model regression estimates of the percentage of an option award that is

exercised in the current month, for certain subsamples of observations. The full sample includes

2.14 million monthly observations associated with 63,244 option grants that are exercised by

19,192 employees in 3,006 individual firms between 1996 and 2013, using records from the

Thomson Reuters Insiders Data file. For a given option award, the percentage of options

exercised is the number exercised in a given month divided by the number of vested options held

at the start of the month. In the first column, we show estimates only for those executives

identified as chief executive officers in the Thomson Reuters database. In the middle column,

we restrict the analysis to those observations in the top quartile of the volatility variable. In the

right column, we restrict the estimation to those observations in the top quartile of the ambiguity

variable. Expected volatility, based on CRSP daily stock price records, and expected ambiguity,

based on TAQ intra-day stock price records, are calculated according to procedures described in

the text. The regression models contain all of the control variables from the model in Table 4

above. t-statistics clustered by person, firm, month and year appear in parentheses below each

coefficient estimate.

Subsample

CEOs

only

25% highest

volatility

25% highest

ambiguity

Intercept

Expected volatility

Expected ambiguity

0.2535

(67.60)

-0.1275

(-2.86)

0.0146

(2.63)

0.2681

(82.37)

-0.0825

(-2.79)

0.0110

(2.47)

0.2576

(72.31)

-0.1854

(-5.13)

0.0146

(3.01)

Year fixed effects

Firm fixed effects

Person fixed effects

Observations

R2

Yes

Yes

Yes

353,556

0.055

Yes

Yes

Yes

535,684

0.073

Yes

Yes

Yes

535,684

0.064